24 PHILOSOPHICAt TRAN^CTIONS. [aNNO 16Q5. 



That the logarithm of the ratio of the geometrical mean to the arithmetical, 

 between 22 and 24, or of i/528 to 23, will be found to be either 

 ^ _L , \ . 4. I I ____! fee or 



1068 ' 1119364, ~ 888215334 ^ 6264878822*8 



J_ . ! . __JL_ &c 



1057 ' 3542796579 659676558485285 



All these series being to be multiplied into 0,4342944819, &c. if you design to 

 make the logarithm of Briggs. But with great advantage in respect of the 

 work, the said 43429448 J 9 &c. is divided by 1057, and the quotient thereof 

 again divided by 3 times the square of 1057, and that quotient again by 4- 

 of that square, and that quotient by ^ thereof, and so on till you have as many 

 figures of the logarithm as you desire. As for example, the logarithm of the 

 geometrical mean between 22 and 24 is found by the logarithms of 2, 3, and 



11, to be 



1 .36 1 3 1 696 1 266906 1 2945009 1 726698O5 



1057 ) 43429 &C. ( 41087462810146814347315886368 



3 in 1117249 ) 41087 &c. ( 12258521544181829460074 



f in 1117249 ) 12258 &c. ( 6583235184376175 



4- in 1 1 17249 ) 65832 &c. ( 4208829765 



■f- in 1117249 ) 42088 &c. (______ 2930 



Summa 1.36l7278360175928788677771 12251 17 



Which is the logarithm of 23 to thirty-two places, and obtained by five divi- 

 sions with very small divisors, all which is much less work than simply multi- 

 plying the series into the said multiplicator 43429 &c. 



Before I pass on to the converse of this problem, or to show how to find the 

 number appertaining to a logarithm assigned, it will be requisite to advertise the 

 reader, tliat there is a small mistake in the aforesaid Mr. James Gregory's Vera 

 Quadratura Circuli et Hyperbolae, published at Padua, Anno 1667, wherein he 

 applies his quadrature of the hyperbola to the making the logarithms: in p. 48 

 he gives the computation of the Lord Napier's logarithm of 10, to 25 places, 

 and finds it 23O2585O92994O45024O1787O instead of 



2302585092994045684017991,* erring in the 18th figure, as I was assured 

 upon my own examination of the number I here give you, and by comparison 

 thereof with the same wrought by another hand, agreeing therewith to 57 of 

 the 60 places. Being desirous to be satisfied how this difference arose, I took 

 the no small trouble of examining Mr. Gregory's work, and at length found 



• This mistake was before noticed by Euclid Speidall, vis. in his Logarithms, published 

 Anno 1688. 



